Chord of Contact

Chord joining the two tangent contact points from external point (x₁, y₁). Same form as the tangent at a point.

Let $P(x_1, y_1)$ be external to $x^2/a^2 - y^2/b^2 = 1$. Tangents from $P$ touch the hyperbola at $A(x_2, y_2)$ and $B(x_3, y_3)$.

Tangent at $A$: $xx_2/a^2 - yy_2/b^2 = 1$. Passes through $P$:

$$\frac{x_1x_2}{a^2} - \frac{y_1y_2}{b^2} = 1 \quad \cdots (1)$$

Tangent at $B$: $xx_3/a^2 - yy_3/b^2 = 1$. Passes through $P$:

$$\frac{x_1x_3}{a^2} - \frac{y_1y_3}{b^2} = 1 \quad \cdots (2)$$

Both say that $A$ and $B$ satisfy:

$$\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$$

This is the chord of contact — the line through both contact points.

The formula is structurally identical across all conics (T = 0), with the specific signs from the conic equation carried through.